### The Extended Centralizer of a Ring Over a Module

R. E. Johnson
1951 Proceedings of the American Mathematical Society
In a recent paper,1 K. Asano gave a new proof of the theorem that a domain of integrity has a right quotient ring if and only if every pair of nonzero elements has a common nonzero right multiple. His method of proof is used in the present work to extend the centralizer of a ring over a module to a system of semi-endomorphisms of the module. From this extension, necessary and sufficient conditions that a ring have a right quotient regular ring are derived. Consider a given ring R, and a given
more » ... ng R, and a given nonzero right i?-moduIe M. Denote by 3JÎ the set of all submodules of M, and by SDÎ* the set of all submodules N oí M having the property that NÍ^N't^O for all nonzero TV'GSOî-Since ME^R*, SDÎ* is not void. It is easily seen that if N and N' are in M*, then N+N' and NC\N' are also in Stt*. Thus {9ÏÏ*; Q, r\, +} is a sublattice of the lattice {W; Q, C\, + }. An i?-homomorphism of N into M, N any element of SO?, is called a semi-endomorphism of M. Thus, thinking of the semi-endomorphism a as a left operator on N, we have a(x+y) =ax+ay and a(xa) = (ax)a for all x, yEN, aER-For convenience, the module N on which a is defined is denoted by Ma. The set of all semi-endomorphisms of M is labeled with 31. Contained in 21 is the usual centralizer of R over M consisting of all a E 21 for which Ma = M. A partial ordering = is defined in 2Í as follows: a^ß if and only if MaÇZMp and ax=ßx for all xEMa. The notation a<ß is used in case a=ß and Maj^Mß. In case ? is a linearly ordered subset of 21, and M' = [)Ma, «GS. the mapping 7 of M' into M defined by 7X = ax whenever x G Ma, a E2, is easily verified to be an element of 21 such that y=a for all aE2-Thus, by Zorn's Lemma, every a oí 21 is contained in a maximal element of 21. Let 58 denote the set of all maximal elements of 21. Obviously the centralizer of R over M is contained in S3. For any /3G33, MßE'HR*-Otherwise there would exist a nonzero NEWl such that Nf~\Mß = 0, and the semi-endomorphism a defined by ax = ßx, x E Mß-, ax = 0, x E N, would exceed ß.